Q. 5 - Calculus

Question

Sketching parametric equations: Sketch the curves defined by the given sets of parametric equations. Indicate the direction of motion on each curve.

$x=sec t$, $y=\tan t$ $t\in \left[-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right]$.

Step-by-Step Solution

Answer

The curve is hyperbola and graph is as follows,

Step 1. Given Information.

The equations are $x=sec t$ and $y=\tan t$ such as $t\in \left[-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right]$.

Step 2. Explanation.

Start by eliminating the parameter. Square the equations for x and y.

$x^2=se{c}^{2}t$

$y^2={\tan }^{2}t$

Apply the Pythagorean identity.  

$$\begin{gathered} se{c}^{2}t-{\tan }^{2}t=1 \\ {x}^2-y^2=1 \end{gathered}$$

The equation represents hyperbola.

Step 3. Calculation.

Make a table to draw the curve.

t	$-\frac{\mathrm{\pi }}{2}$	$-\frac{\mathrm{\pi }}{4}$	0	$\frac{\mathrm{\pi }}{4}$	$\frac{\mathrm{\pi }}{2}$
$x=sec t$	$\infty$	$\sqrt{2}$	1	$\sqrt{2}$	$\infty$
$y=\tan t$	0	-1	0	1	0

Step 4. Graph.

The required graph is as follows,